Fluid Mechanics Course Code: CHPE207 and CIVL213 Lecturer: Dr Mustafa Saleh Nasser Office Number: 5D-44 E-mail: mustafa.nasser@uniwa.edu.om Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dr Mustafa Nasser

Dimensions and Units In fluid mechanics we must describe various fluid characteristics in terms of certain basic quantities such as length, time and mass A dimension is the measure by which a physical variable is expressed qualitatively, i.e. length is a dimension associated with distance, width, height, displacement. Basic dimensions: Length, L (or primary quantities) Time, T Mass, M Temperature, Q We can derive any secondary quantity from the primary quantities i.e. Force = (mass) x (acceleration) : F = M L T-2 A unit is a particular way of attaching a number to the qualitative dimension: Systems of units can vary from country to country, but dimensions do not Dr Mustafa Nasser

Dr Mustafa Nasser

Units of Force – E system To make Newton’s law dimensionally consistent we must include a dimensional proportionality constant: where Dr Mustafa Nasser

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3 Dr Mustafa Nasser 14 14

Density for Gasses For gases: strong function of T and P from ideal gas law: r = P M/R T (can you drive this) where R = universal gas constant, M=mol. weight R= 8.314 J/(g-mole K)=0.08314 (liter bar)/(g-mole K)= 0.08206 (liter atm)/(g-mole K)=1.987 (cal)/(g-mole K)= 10.73 (psia ft3)/(lb-mole °R)=0.7302 (atm ft3)/(lb-mole °R) Dr Mustafa Nasser

Pressure Pascal’s Laws Pascals’ laws: Pressure is defined as the amount of force exerted on a unit area of a substance: P = F / A Pascal’s Laws Pascals’ laws: Pressure acts uniformly in all directions on a small volume (point) of a fluid In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary – it is a normal force. Dr Mustafa Nasser

Direction of fluid pressure on boundaries Furnace duct Pipe or tube Heat exchanger Pressure is due to a Normal Force (acting perpendicular to the surface) It is also called a Surface Force Dam Dr Mustafa Nasser

Absolute and Gauge Pressure Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere. Dr Mustafa Nasser

Units for Pressure Unit Definition or Relationship 1 pascal (Pa) 1 kg m-1 s-2 1 bar 1 x 105 Pa 1 atmosphere (atm) 101,325 Pa 1 torr 1 / 760 atm 760 mm Hg 1 atm 14.696 pounds per sq. in. (psi) Dr Mustafa Nasser

Pressure distribution for a fluid at rest We will determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity The sum of the forces acting on the fluid must equal zero Consider an infinitesimal rectangular fluid element of dimensions Dx, Dy, Dz z y x Dr Mustafa Nasser

Pressure distribution for a fluid at rest Let Pz and Pz+Dz denote the pressures at the base and top of the cube, where the elevations are z and z+Dz respectively. Force at base of cube: Pz A=Pz (Dx Dy) Force at top of cube: Pz+Dz A= Pz+Dz (Dx Dy) Force due to gravity: m g=r V g = r (Dx Dy Dz) g A force balance in the z direction gives: Fg For an infinitesimal element (Dz0)  Dr Mustafa Nasser

Variation of pressure with elevation

General variation of pressure in a static fluid due to gravity

General variation of pressure in a static fluid due to gravity for vertical direction: q = 0

Equality of pressure at the same level in a static fluid

Incompressible fluid By using gauge pressures we can simply write: Liquids are incompressible i.e. their density is assumed to be constant. When we have a liquid with a free surface the pressure P at any depth below the free surface is: where Po is the pressure at the free surface (Po=Patm) and h = zfree surface - z By using gauge pressures we can simply write: Dr Mustafa Nasser

Equality of pressure at the same level in a continuous fluid

Examples SG= 13.6 Dr Mustafa Nasser

Solution: 1.1 1.2 Dr Mustafa Nasser

1.3 Dr Mustafa Nasser

Calculate the weight of a reservoir of oil if it has a mass of 825 kg. We have

If the reservoir from previous Example has a volume of 0 If the reservoir from previous Example has a volume of 0.917m3, compute the density, the specific weight, and the specific gravity of the oil.

3.7 Pressure expressed as the Height of a Column of Liquid Then we can use this as a conversion factor,

Pressure Gages and Transducers For those situations where only a visual indication is needed at the site where the pressure is being measured, a pressure gage is used. In other cases there is a need to measure pressure at one point and display the value at another. The general term for such a device is pressure transducer, meaning that the sensed pressure causes an electrical signal to be generated that can be transmitted to a remote location such as a central control station where it is displayed digitally.

Below Figs shows the Bourdon tube pressure gage. Pressure Gages Below Figs shows the Bourdon tube pressure gage.

Pressure Gages The pressure to be measured is applied to the inside of a flattened tube, which is normally shaped as a segment of a circle or a spiral. The increased pressure inside the tube causes it to be straightened somewhat. The movement of the end of the tube is transmitted through a linkage that causes a pointer to rotate.

Pressure Gages Figure below shows a pressure gage using an actuation means called Magnehelic pressure gage.

3.8.2 Strain Gage Pressure Transducer Fig 3.17 shows the strain gage pressure transducer and indicator. The pressure to be measured is introduced through the pressure port and acts on a diaphragm to which foil strain gages are bonded. As the strain gages sense the deformation of the diaphragm, their resistance changes. The readout device is typically a digital voltmeter, calibrated in pressure units.

Strain Gage Pressure Transducer

LVST-Type Pressure Transducer A linear variable differential transformer (LVDT) is composed of a cylindrical electric coil with a movable rod-shaped core. As the core moves along the axis of the coil, a voltage change occurs in relation to the position of the core. This type of transducer is applied to pressure measurement by attaching the core rod to a flexible diaphragm. Fig shows the Linear variable differential transformer (LVDT)-type pressure transducer.

LVST-Type Pressure Transducer

Piezoelectric Pressure Transducer Certain crystals, such as quartz and barium titanate, exhibit a piezoelectric effect, in which the electrical charge across the crystal varies with stress in the crystal. Causing a pressure to exert a force, either directly or indirectly, on the crystal leads to a voltage change related to the pressure change. Fig shows the digital pressure gage.

Piezoelectric Pressure Transducer

Buoyancy Archimedes Principle Laws of buoyancy discovered by Archimedes: A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces A floating body displaces its own weight in the fluid in which it floats Free liquid surface F1 h1 The upper surface of the body is subjected to a smaller force than the lower surface  A net force is acting upwards H h2 F2 Dr Mustafa Nasser

Buoyancy where r = the fluid density The net force due to pressure in the vertical direction is: FB = F2- F1 = (Pbottom - Ptop) (DxDy) The pressure difference is: Pbottom – Ptop = r g (h2-h1) = r g H FB = r g H (DxDy) Thus the buoyant force is: FB = r g V where r = the fluid density If the fluid density is greater than the average density of the object, the object floats. If less, the object sinks Dr Mustafa Nasser

Example Consider a solid cube of dimensions 1ft x 1ft x 1ft (=0.305m x 0.305m x 0.305m). Its top surface is 10 ft (=3.05 m) below the surface of the water. The density of water is rf=1000 kg/m3. Consider two cases: The cube is made of cork (rB=160.2 kg/m3) b) The cube is made of steel (rB=7849 kg/m3) In what direction does the body tend to move? Dr Mustafa Nasser

Dr Mustafa Nasser

Shear in Different Fluids Shear-stress relations for different types of fluids Newtonian fluids: linear relationship Slope of line (coefficient of proportionality) is “viscosity”

Dr Mustafa Nasser

Viscosity The constant of proportionality is designated by the Greek symbol  (mu) and is called the absolute viscosity, dynamic viscosity, or simply the viscosity of the fluid. The viscosity depends on the particular fluid, and for a particular fluid the viscosity is also dependent on temperature.

Viscosity and Temperature 1/3 For fluids, the viscosity decreases with an increase in temperature. For gases, an increase in temperature causes an increase in viscosity. WHY? molecular structure.

Fluid Properties Examples Explain why the viscosity of a liquid decreases while that of a gas increases with a temperature rise. The following is a table of measurement for a fluid at constant temperature. Determine the dynamic viscosity of the fluid. du/dy (rad s-1) 0.00 0.20 0.40 0.60 0.80 t (N m-2) 0.01 1.90 3.10 4.00 Dr Mustafa Nasser 52

Dr Mustafa Nasser 53 Using Newton's law of viscocity where m is the viscosity. So viscosity is the gradient of a graph of shear stress against vellocity gradient of the above data, or Plot the data as a graph: Calculate the gradient for each section of the line du/dy (s-1) 0.0 0.2 0.4 0.6 0.8 t (N m-2) 1.0 1.9 3.1 4.0 Gradient - 5.0 4.75 5.17 Thus the mean gradient = viscosity = 4.98 N s / m2 Dr Mustafa Nasser 53

Example 2: 5. 6m3 of oil weighs 46 800 N Example 2: 5.6m3 of oil weighs 46 800 N. Find its mass density, and relative density. Solution 2: Weight 46 800 = mg Mass m = 46 800 / 9.81 = 4770.6 kg Mass density r = Mass / volume = 4770.6 / 5.6 = 852 kg/m3 Relative density  Dr Mustafa Nasser

Example 3 The velocity distribution of a viscous liquid (dynamic viscosity m = 0.9 Ns/m2) flowing over a fixed plate is given by u = 0.68y - y2 (u is velocity in m/s and y is the distance from the plate in m). What are the shear stresses at the plate surface and at y=0.34m? Dr Mustafa Nasser 55

Calculate the shear stress at the plate face At the plate face y = 0m, Calculate the shear stress at the plate face At y = 0.34m, As the velocity gradient is zero at y=0.34 then the shear stress must also be zero. Dr Mustafa Nasser 56

Example 4 In a fluid the velocity measured at a distance of 75mm from the boundary is 1.125m/s. The fluid has absolute viscosity 0.048 Pa s and relative density 0.913. What is the velocity gradient and shear stress at the boundary assuming a linear velocity distribution. m = 0.048 Pa s g = 0.913 Dr Mustafa Nasser 57

Newtonian and Non-Newtonian Fluid Fluids for which the shearing stress is linearly related to the rate of shearing strain are designated as Newtonian fluids Most common fluids such as water, air, and gasoline are Newtonian fluid under normal conditions. Fluids for which the shearing stress is not linearly related to the rate of shearing strain are designated as non-Newtonian fluids.

Non-Newtonian Fluids

Non-Newtonian Fluids Newtonian Fluid Non-Newtonian Fluid η is the apparent viscosity and is not constant for non-Newtonian fluids.

η - Apparent Viscosity The shear rate dependence of η categorizes non-Newtonian fluids into several types. Power Law Fluids: Pseudoplastic – η (viscosity) decreases as shear rate increases (shear rate thinning) Dilatant – η (viscosity) increases as shear rate increases (shear rate thickening) Bingham Plastics: η depends on a critical shear stress (t0) and then becomes constant

Non-Newtonian Fluids Bingham Plastic: sludge, paint, blood, ketchup Pseudoplastic: latex, paper pulp, clay solns. Newtonian Dilatant: quicksand

non-Newtonian Fluids 1/2 Shear thinning fluids: The viscosity decreases with increasing shear rate – the harder the fluid is sheared, the less viscous it becomes. Many colloidal suspensions and polymer solutions are shear thinning. Latex paint is example.

non-Newtonian Fluids 2/2 Shear thickening fluids: The viscosity increases with increasing shear rate – the harder the fluid is sheared, the more viscous it becomes. Water-corn starch mixture water-sand mixture are examples. Bingham plastic: neither a fluid nor a solid. Such material can withstand a finite shear stress without motion, but once the yield stress is exceeded it flows like a fluid. Toothpaste and mayonnaise are common examples.